cherenkov radiation energy loss of runaway...

Cherenkov Radiation energy loss of runaway electronsChang Liu, Dylan Brennan, Amitava BhattacharjeePrinceton University, PPPLAllen BoozerColumbia University

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Runaway Electron Meeting 2016Pertuis, France

Jun 7 2016

Outline•  Introduction to Cherenkov radiation in plasma

•  Lenard-Balescu collision operator

• Cherenkov radiation in unmagnetized plasma• Radiative energy loss• Correction to Coulomb logarithm

• Cherenkov radiation in magnetized plasma•  Transit-time magnetic pumping (TTMP) momentum loss

•  Summary

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•  Summary

3

•  Cherenkov radiation is emitted when a charged particle passed through a dielectric medium at a speed greater than wave phase velocity in that medium.

•  The condition for Cherenkov radiation is n>1/β (n=kc/ω, β=v/c).•  For plasma waves and runaway electrons (β~1), this condition is easy to

satisfy.

Introduction: Cherenkov radiation

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Introduction: Lenard-Balescu collision operator and Cherenkov radiation in plasmas

•  The polarization drag originates from the polarization of the plasma medium due to the electric field of a test particle, which is related to the Cherenkov radiation energy loss.•  For ω/k~vth, the excited mode is strongly Landau damped and the energy

is absorbed by bulk electrons.•  For ω/k≫vth, the mode is weakly damped (mainly through collisions), and

energy gets radiated away.

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C[ f ]= ∂

∂v⋅ q

m⎛⎝⎜

⎞⎠⎟ E

p f −D ⋅ ∂ f∂v

⎡⎣⎢

⎤⎦⎥

Ep: Polarization drag electric fieldD: Diffusion from electric field fluctuation

D = 1

2qm

⎛⎝⎜

⎞⎠⎟2

dk δEδE (k,ω = k ⋅v)∫

Test particle superposition principle





•  Summary

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Electron waves in unmagnetized plasma

•  In unmagnetized plasma, the electron waves have two branches (ignoring ion effects and thermal correction)

• 

• 

•  For Cherenkov radiation, only Langmuir wave is possible (n>1).

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Langmuir wave: ω 2 =ω p2

Electronmagnetic wave: ω 2 = k2c2 +ω p2

0.1 0.5 1 5 100

1

2

3

4

ω/ωpen

The energy loss from Cherenkov radiation

•  To solve the energy loss due to Cherenkov radiation, one can calculate the excited electric field from a single moving electron in the medium.

•  The current from a single moving electron

•  The time-averaged electric field felt by the moving electron

•  The energy loss is then Ep·j.

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Ampere's lawFaraday's law

⎫⎬⎪

⎭⎪→ k × k ×E+ ω

c⎛⎝⎜

⎞⎠⎟

2

ε ⋅E = − 4πiωc2

j

j(x,t) = −evδ (x − x0 − vt)→ j(k,ω ) = −evδ (ω − k ⋅v)exp(ik ⋅x0 )

Ep = d 3k dω E(k,ω )exp −ik(x0 + vt)+ iωt[ ]∫∫ t

Cherenkov resonance

Cherenkov radiation energy loss in unmagnetized plasma•  For unmagnetized plasma

•  Assume electron is moving along z direction,

•  The principal value of the integral is imaginary, which will not contribute to the energy loss. •  However, the residues (which correspond to the modes that satisfy the

Cherenkov radiation condition) will give real contribution and energy loss.

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ε = 1−

ω p2

ω 2⎛

⎝⎜⎞

⎠⎟1 = ε1

Ezp = d 3kdω

4πi evc2 ε −k!2c2

ω 2⎛⎝⎜

⎞⎠⎟

ωε ε − k2c2

ω 2⎛⎝⎜

⎞⎠⎟

∫ δ (ω − k ⋅v), k! =k ⋅vv

Correction to Coulomb logarithm due to Cherenkov radiation loss•  Using the residue theorem to calculate the integral

•  Note that we use the cold plasma approximation, so we can choose kmax=1/λD to ensure thermal effect is not important.

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Ezp =

eω p2

v2sinθdθcosθ

σ (θ ), σ (θ ) =1, if cosθ >ω p / kmaxv

0, if cosθ >ω p / kmaxv

⎧⎨⎪

⎩⎪∫

=eω p

2

v2ln kmaxv

ω pθ Is the wave emittance angle

Correction to lnΛ for the drag force

•  Recall that the drag force in the Landau collision operator due to binary collisions can be written as

•  Cherenkov radiation energy loss gives a correction to lnΛ

•  In DIII-D QRE experiments, for highly relativistic runaway electrons, this correction is about 20%.

•  For electron starting to run away (v≪c), this correction is small.

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Edrag =eω p

2

v2lnΛ, Λ = bmax

bmin

bmax = λD , bmin = maxeαeβmαβu

2 ,!

2mαβu⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

lnΛ→ lnΛ + ln vvth

⎛⎝⎜

⎞⎠⎟





•  Summary

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Cherenkov radiation in magnetized plasma

•  The dielectric tensor and the dispersion relation in magnetized plasma (ignoring ions and thermal effects)

•  The allowed wave branches for Cherenkov radiation are: Langmuir wave, extraordinary-electron (EXEL) wave, and upper-hybrid wave (electrostatic Bernstein wave).

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0.1 0.5 1 5 100

1

2

3

4

5

6

7

ω/ωpen

ωce

θ=5°

n=1/cosθ

ε =

1−ω pe

2

ω 2 −ω ce2 −i

ω ceω

ω pe2

ω 2 −ω ce2 0

iω ceω

ω pe2

ω 2 −ω ce2 1−

ω pe2

ω 2 −ω ce2 0

0 0 1−ω pe

2

ω 2

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

G.I. Pokol, A. Kómár, A. Budai, A. Stahl, and T. Fülöp, Phys. Plasmas 21, 102503 (2014).

0.1 0.5 1 5 100

1

2

3

4

5

6

7

ω/ωpen

Cherenkov radiation in magnetized plasma

•  The dielectric tensor and the dispersion relation in magnetized plasma (ignoring ions and thermal effects)

•  The allowed wave branches for Cherenkov radiation are: Langmuir wave, extraordinary-electron (EXEL) wave, and upper-hybrid wave (electrostatic Bernstein wave).

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ωUH

θ=80°

n=1/cosθ

ε =

1−ω pe

2

ω 2 −ω ce2 −i

ω ceω

ω pe2

ω 2 −ω ce2 0

iω ceω

ω pe2

ω 2 −ω ce2 1−

ω pe2

ω 2 −ω ce2 0

0 0 1−ω pe

2

ω 2

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟

G.I. Pokol, A. Kómár, A. Budai, A. Stahl, and T. Fülöp, Phys. Plasmas 21, 102503 (2014).

Wave frequency for Cherenkov radiation

•  For a given emittance angle of k, one can calculate the wave frequency that satisfies the Cherenkov radiation condition.

•  For ωce~ωpe, one emittance angle gives one solution of ω, which shifts from ωpe to ωUH.

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0.2 0.4 0.6 0.8 1.0 1.2 1.4

1.0

1.5

2.0

2.5

3.0

3.5

θ

ω/ω

pe

ω1=ωpe

ω2=ωUH

ωce=3ωpe

0.1 0.5 1 5 100

1

2

3

4

ω/ωpe

n

θ=0.5

n=1/cosθ


•  For ωce≫ωpe, there are 3 branches of ω roots, which all coexist in a range of emittance angle.

Wave frequency for Cherenkov radiation (cont’d)

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0.2 0.4 0.6 0.8 1.0

2

4

6

8

10

θ

ω/ω

pe

0.1 0.5 1 5 100

1

2

3

4

ω/ωpe

nω1=ωpe

ω2=ωUH

ωce=10ωpe

θ=0.2

n=1/cosθ


•  For ωce≫ωpe, there are 3 branches of ω roots, which all coexist in a range of emittance angle.

0.5 1 5 10

0.96

0.98

1.00

1.02

1.04

1.06

1.08

1.10

ω/ωpe

n

Wave frequency for Cherenkov radiation (cont’d)

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0.2 0.4 0.6 0.8 1.0

2

4

6

8

10

θ

ω/ω

pe

ω1=ωpe

ω2=ωUH

ωce=10ωpe

θ=0.2

n=1/cosθ

Cherenkov radiation energy loss in magnetized plasma•  We can use the same method to calculate the Cherenkov radiation energy

loss.•  Assume that electron is moving perfectly along B field (no v⊥)

•  Although the radiation loss from small emittance angle (Langmuir) and large angle (Upper-hybrid) stays the same, intermediate angle (EXEL wave) gives additional energy loss.

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

5

10

15

20

θ

A(θ)

ωce/ωpe=0.2

ωce/ωpe=3

ωce/ωpe=10 Ezp =

eω pe2

v2sinθdθcosθ

A(θ )∫

Effects of 3 modes on Cerenkov radiation energy loss

•  All the 3 modes contributed to the Cherenkov radiation energy loss•  For ωce≫ωpe, the radiation emittance is strongly peaked at certain

angle.

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Total

ωce/ωpe=10

Cherenkov radiation energy loss in magnetized plasma (cont’d)•  The Coulomb logarithm in the drag force has a correction

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Δ2 =x(a1x + a0 )(x2 + b1x + b0 )

, x = ω ceω pe

a1 = 2.05,a0 = 9.51b1 = 1.63,b0 = 26.23

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

ωce/ωpe

Δ 2

lnΛ→ lnΛ + ln vvth

⎛⎝⎜

⎞⎠⎟+ Δ2

Transit-time magnetic pumping (TTMP) associated with Cherenkov radiation•  Now we consider runaway electrons with finite v⊥.•  In addition to electric field, gradient of magnetic field can also cause

momentum loss in parallel direction

•  Note that for electron moving near the speed of light, the effect from the electric force (E) and Lorentz force (v×B/c) can be of similar order.

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Magnetic pumping: FTTMP = −µ∇Bz

TTMP momentum loss•  From Faraday’s law, Bz=kx cEy/ω. So the TTMP force can be calculated similarly from Ey.

•  The ratio of TTMP over polarization electric force

•  For high energy regime where synchrotron radiation energy loss dominates, v⊥/c~1/γ, the effect of TTMP is small.•  For high energy runaway electrons, synchrotron and Bremsstrahlung radiation dominate

•  With fast pitch angle scattering mechanism (different from collisional scattering) such as whistler wave, the effect can be more important.

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FTTMP =µω cec2ω pe

ω pe2

v2ln v

vth

⎛⎝⎜

⎞⎠⎟− 121− vth

v⎛⎝⎜

⎞⎠⎟ + Δ3

⎡

⎣⎢

⎤

⎦⎥

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

ωce/ωpe

Δ 3

FTTMPeE p

≈ γ v⊥c

⎛⎝⎜

⎞⎠⎟2





•  Summary

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Summary•  Cherenkov radiation causes runaway electrons to lose energy, which can be

described by adding a correction to the Coulomb logarithm in the drag force.•  The correction is about 20% compared to collisional force in DIII-D QRE

experiment

•  Magnetic field enhances Cherenkov radiation and energy loss.•  For runaway electrons with finite v⊥, TTMP will cause momentum loss on

parallel direction.

•  Future work:•  Including thermal effect in the plasma wave description•  Study the spiral orbit particle rather than straight orbit•  Study the electric field fluctuation given by Cherenkov radiation

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Questions for discussion1.  Can we detect the Cherenkov radiation from runaway electrons?

2.  What is the correct physics interpretation of Cherenkov resonance condition when collisional damping is considered?

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ω = k ⋅vexp(ikx − iωt)

Imω < 0,Im k < 0 Wave is damping in time, but growing in space?Imω > 0,Im k > 0 Wave is damping in space, but growing in time?

⎧⎨⎪

⎩⎪

B. J. Tobias et al., Rev. Sci. Instrum. 83, 10E329 (2012)

cherenkov radiation energy loss of runaway...

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